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        <p><a href="https://xgboost.readthedocs.io/en/latest/tutorials/model.html" target="_blank" rel="noopener">Doc</a><br><a href="https://homes.cs.washington.edu/~tqchen/pdf/BoostedTree.pdf" target="_blank" rel="noopener">Slides</a></p>
<h1 id="为何要推导出目标函数而不是直接增加树"><a href="#为何要推导出目标函数而不是直接增加树" class="headerlink" title="为何要推导出目标函数而不是直接增加树"></a>为何要推导出目标函数而不是直接增加树</h1><p><img src="http://i.imgur.com/quPhp1K.png" alt="Objective function"></p>
<ul>
<li>理论上：搞清楚learning的目的，以及其收敛性。</li>
<li>工程上：<ul>
<li>gi和hi是对loss function的一次、二次导</li>
<li>目标函数以及整个学习过程只依赖于gi和hi</li>
<li>可以根据实际问题，自定义loss function</li>
</ul>
</li>
</ul>
<h1 id="Summary"><a href="#Summary" class="headerlink" title="Summary"></a>Summary</h1><p><img src="http://i.imgur.com/L7PhJwO.png" alt="Summary"></p>
<h1 id="原理"><a href="#原理" class="headerlink" title="原理"></a>原理</h1><h2 id="损失函数"><a href="#损失函数" class="headerlink" title="损失函数"></a>损失函数</h2><p>$$ \text{obj} = \sum_{i=1}^n l(y_i, \hat{y}<em>i^{(t)}) + \sum</em>{i=1}^t\Omega(f_i)$$<br>l为loss，\ \Omega \ 为正则项</p>
<ul>
<li>loss：采用加法策略，第t颗树时：<br>$$ \hat{y}_i^{(0)} = 0 $$<br>$$ \hat{y}_i^{(1)} = f_1(x_i) = \hat{y}_i^{(0)} + f_1(x_i) $$<br>$$ \hat{y}_i^{(2)} = f_1(x_i) + f_2(x_i)= \hat{y}_i^{(1)} + f_2(x_i) $$<br>$$ \dots $$<br>$$ \hat{y}<em>i^{(t)} = \sum</em>{k=1}^t f_k(x_i)= \hat{y}_i^{(t-1)} + f_t(x_i) $$<br>在添加第t颗树时，需要优化的目标函数为：<br>$$ \sum_{i=1}^n [g_i f_t(x_i) + \frac{1}{2} h_i f_t^2(x_i)] + \Omega(f_t) $$<br>其中h和f：<br>$$ g_i = \partial_{\hat{y}_i^{(t-1)}} l(y_i, \hat{y}_i^{(t-1)}) $$<br>$$ h_i = \partial_{\hat{y}_i^{(t-1)}}^2 l(y_i, \hat{y}_i^{(t-1)}) $$<br>note: 是对谁的导</li>
<li>正则项：复杂度：<br>$$ \Omega(f) = \gamma T + \frac{1}{2}\lambda \sum_{j=1}^T w_j^2 $$<br>其中w是叶子上的score vector，T是叶子数量</li>
</ul>
<h2 id="DART-Booster"><a href="#DART-Booster" class="headerlink" title="DART Booster"></a>DART Booster</h2><p>为了解决过拟合，会随机drop trees:</p>
<ul>
<li>训练速度可能慢于gbtree</li>
<li>由于随机性，早停可能不稳定</li>
</ul>
<h1 id="特性"><a href="#特性" class="headerlink" title="特性"></a>特性</h1><h2 id="Monotonic-Constraints单调性限制"><a href="#Monotonic-Constraints单调性限制" class="headerlink" title="Monotonic Constraints单调性限制"></a>Monotonic Constraints单调性限制</h2><ul>
<li><p>一个可选特性:<br>会限制模型的结果按照某个特征 单调的进行增减</p>
<p>也就是说可以降低模型对数据的敏感度，如果明确已知某个特征与预测结果呈单调关系时，那在生成模型的时候就会跟特征数据的单调性有关。</p>
</li>
</ul>
<h2 id="Feature-Interaction-Constraints单调性限制"><a href="#Feature-Interaction-Constraints单调性限制" class="headerlink" title="Feature Interaction Constraints单调性限制"></a>Feature Interaction Constraints单调性限制</h2><ul>
<li><p>一个可选特性：<br>不用时，在tree生成的时候，一棵树上的节点会无限制地选用多个特征</p>
<p>设置此特性时，可以规定，哪些特征可以有interaction（一般独立变量之间可以interaction，非独立变量的话可能会引入噪声）</p>
</li>
<li>好处：<ul>
<li>预测时更小的噪声</li>
<li>对模型更好地控制</li>
</ul>
</li>
</ul>
<h2 id="Instance-Weight-File"><a href="#Instance-Weight-File" class="headerlink" title="Instance Weight File"></a>Instance Weight File</h2><ul>
<li>规定了模型训练时data中每一条instance的权重</li>
<li>有些instance质量较差，或与前一示例相比变化不大，所以可以调节其所占权重</li>
</ul>
<h1 id="调参"><a href="#调参" class="headerlink" title="调参"></a>调参</h1><h2 id="Overfitting"><a href="#Overfitting" class="headerlink" title="Overfitting"></a>Overfitting</h2><p>与overfitting有关的参数：</p>
<ul>
<li>直接控制模型复杂度：max_depth, min_child_weight and gamma.</li>
<li>增加模型随机性以使得模型对噪声有更强的鲁棒性：<ul>
<li>subsample and colsample_bytree. </li>
<li>Reduce stepsize eta. Remember to increase num_round when you do so.</li>
</ul>
</li>
</ul>
<h2 id="Imbalanced-Dataset"><a href="#Imbalanced-Dataset" class="headerlink" title="Imbalanced Dataset"></a>Imbalanced Dataset</h2><ul>
<li>只关注测量指标的大小<ul>
<li>平衡数据集 via scale_pos_weight</li>
<li>使用AUC作为metric</li>
</ul>
</li>
<li>关注预测正确的概率<ul>
<li>此时不能re-balance数据集</li>
<li>Set parameter max_delta_step to a finite number (say 1) to help convergence</li>
</ul>
</li>
</ul>

      
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              <div class="post-toc-content"><ol class="nav"><li class="nav-item nav-level-1"><a class="nav-link" href="#为何要推导出目标函数而不是直接增加树"><span class="nav-number">1.</span> <span class="nav-text">为何要推导出目标函数而不是直接增加树</span></a></li><li class="nav-item nav-level-1"><a class="nav-link" href="#Summary"><span class="nav-number">2.</span> <span class="nav-text">Summary</span></a></li><li class="nav-item nav-level-1"><a class="nav-link" href="#原理"><span class="nav-number">3.</span> <span class="nav-text">原理</span></a><ol class="nav-child"><li class="nav-item nav-level-2"><a class="nav-link" href="#损失函数"><span class="nav-number">3.1.</span> <span class="nav-text">损失函数</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#DART-Booster"><span class="nav-number">3.2.</span> <span class="nav-text">DART Booster</span></a></li></ol></li><li class="nav-item nav-level-1"><a class="nav-link" href="#特性"><span class="nav-number">4.</span> <span class="nav-text">特性</span></a><ol class="nav-child"><li class="nav-item nav-level-2"><a class="nav-link" href="#Monotonic-Constraints单调性限制"><span class="nav-number">4.1.</span> <span class="nav-text">Monotonic Constraints单调性限制</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#Feature-Interaction-Constraints单调性限制"><span class="nav-number">4.2.</span> <span class="nav-text">Feature Interaction Constraints单调性限制</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#Instance-Weight-File"><span class="nav-number">4.3.</span> <span class="nav-text">Instance Weight File</span></a></li></ol></li><li class="nav-item nav-level-1"><a class="nav-link" href="#调参"><span class="nav-number">5.</span> <span class="nav-text">调参</span></a><ol class="nav-child"><li class="nav-item nav-level-2"><a class="nav-link" href="#Overfitting"><span class="nav-number">5.1.</span> <span class="nav-text">Overfitting</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#Imbalanced-Dataset"><span class="nav-number">5.2.</span> <span class="nav-text">Imbalanced Dataset</span></a></li></ol></li></ol></div>
            

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  <script type="text/javascript">
    // Popup Window;
    var isfetched = false;
    var isXml = true;
    // Search DB path;
    var search_path = "search.xml";
    if (search_path.length === 0) {
      search_path = "search.xml";
    } else if (/json$/i.test(search_path)) {
      isXml = false;
    }
    var path = "/" + search_path;
    // monitor main search box;

    var onPopupClose = function (e) {
      $('.popup').hide();
      $('#local-search-input').val('');
      $('.search-result-list').remove();
      $('#no-result').remove();
      $(".local-search-pop-overlay").remove();
      $('body').css('overflow', '');
    }

    function proceedsearch() {
      $("body")
        .append('<div class="search-popup-overlay local-search-pop-overlay"></div>')
        .css('overflow', 'hidden');
      $('.search-popup-overlay').click(onPopupClose);
      $('.popup').toggle();
      var $localSearchInput = $('#local-search-input');
      $localSearchInput.attr("autocapitalize", "none");
      $localSearchInput.attr("autocorrect", "off");
      $localSearchInput.focus();
    }

    // search function;
    var searchFunc = function(path, search_id, content_id) {
      'use strict';

      // start loading animation
      $("body")
        .append('<div class="search-popup-overlay local-search-pop-overlay">' +
          '<div id="search-loading-icon">' +
          '<i class="fa fa-spinner fa-pulse fa-5x fa-fw"></i>' +
          '</div>' +
          '</div>')
        .css('overflow', 'hidden');
      $("#search-loading-icon").css('margin', '20% auto 0 auto').css('text-align', 'center');

      $.ajax({
        url: path,
        dataType: isXml ? "xml" : "json",
        async: true,
        success: function(res) {
          // get the contents from search data
          isfetched = true;
          $('.popup').detach().appendTo('.header-inner');
          var datas = isXml ? $("entry", res).map(function() {
            return {
              title: $("title", this).text(),
              content: $("content",this).text(),
              url: $("url" , this).text()
            };
          }).get() : res;
          var input = document.getElementById(search_id);
          var resultContent = document.getElementById(content_id);
          var inputEventFunction = function() {
            var searchText = input.value.trim().toLowerCase();
            var keywords = searchText.split(/[\s\-]+/);
            if (keywords.length > 1) {
              keywords.push(searchText);
            }
            var resultItems = [];
            if (searchText.length > 0) {
              // perform local searching
              datas.forEach(function(data) {
                var isMatch = false;
                var hitCount = 0;
                var searchTextCount = 0;
                var title = data.title.trim();
                var titleInLowerCase = title.toLowerCase();
                var content = data.content.trim().replace(/<[^>]+>/g,"");
                var contentInLowerCase = content.toLowerCase();
                var articleUrl = decodeURIComponent(data.url);
                var indexOfTitle = [];
                var indexOfContent = [];
                // only match articles with not empty titles
                if(title != '') {
                  keywords.forEach(function(keyword) {
                    function getIndexByWord(word, text, caseSensitive) {
                      var wordLen = word.length;
                      if (wordLen === 0) {
                        return [];
                      }
                      var startPosition = 0, position = [], index = [];
                      if (!caseSensitive) {
                        text = text.toLowerCase();
                        word = word.toLowerCase();
                      }
                      while ((position = text.indexOf(word, startPosition)) > -1) {
                        index.push({position: position, word: word});
                        startPosition = position + wordLen;
                      }
                      return index;
                    }

                    indexOfTitle = indexOfTitle.concat(getIndexByWord(keyword, titleInLowerCase, false));
                    indexOfContent = indexOfContent.concat(getIndexByWord(keyword, contentInLowerCase, false));
                  });
                  if (indexOfTitle.length > 0 || indexOfContent.length > 0) {
                    isMatch = true;
                    hitCount = indexOfTitle.length + indexOfContent.length;
                  }
                }

                // show search results

                if (isMatch) {
                  // sort index by position of keyword

                  [indexOfTitle, indexOfContent].forEach(function (index) {
                    index.sort(function (itemLeft, itemRight) {
                      if (itemRight.position !== itemLeft.position) {
                        return itemRight.position - itemLeft.position;
                      } else {
                        return itemLeft.word.length - itemRight.word.length;
                      }
                    });
                  });

                  // merge hits into slices

                  function mergeIntoSlice(text, start, end, index) {
                    var item = index[index.length - 1];
                    var position = item.position;
                    var word = item.word;
                    var hits = [];
                    var searchTextCountInSlice = 0;
                    while (position + word.length <= end && index.length != 0) {
                      if (word === searchText) {
                        searchTextCountInSlice++;
                      }
                      hits.push({position: position, length: word.length});
                      var wordEnd = position + word.length;

                      // move to next position of hit

                      index.pop();
                      while (index.length != 0) {
                        item = index[index.length - 1];
                        position = item.position;
                        word = item.word;
                        if (wordEnd > position) {
                          index.pop();
                        } else {
                          break;
                        }
                      }
                    }
                    searchTextCount += searchTextCountInSlice;
                    return {
                      hits: hits,
                      start: start,
                      end: end,
                      searchTextCount: searchTextCountInSlice
                    };
                  }

                  var slicesOfTitle = [];
                  if (indexOfTitle.length != 0) {
                    slicesOfTitle.push(mergeIntoSlice(title, 0, title.length, indexOfTitle));
                  }

                  var slicesOfContent = [];
                  while (indexOfContent.length != 0) {
                    var item = indexOfContent[indexOfContent.length - 1];
                    var position = item.position;
                    var word = item.word;
                    // cut out 100 characters
                    var start = position - 20;
                    var end = position + 80;
                    if(start < 0){
                      start = 0;
                    }
                    if (end < position + word.length) {
                      end = position + word.length;
                    }
                    if(end > content.length){
                      end = content.length;
                    }
                    slicesOfContent.push(mergeIntoSlice(content, start, end, indexOfContent));
                  }

                  // sort slices in content by search text's count and hits' count

                  slicesOfContent.sort(function (sliceLeft, sliceRight) {
                    if (sliceLeft.searchTextCount !== sliceRight.searchTextCount) {
                      return sliceRight.searchTextCount - sliceLeft.searchTextCount;
                    } else if (sliceLeft.hits.length !== sliceRight.hits.length) {
                      return sliceRight.hits.length - sliceLeft.hits.length;
                    } else {
                      return sliceLeft.start - sliceRight.start;
                    }
                  });

                  // select top N slices in content

                  var upperBound = parseInt('1');
                  if (upperBound >= 0) {
                    slicesOfContent = slicesOfContent.slice(0, upperBound);
                  }

                  // highlight title and content

                  function highlightKeyword(text, slice) {
                    var result = '';
                    var prevEnd = slice.start;
                    slice.hits.forEach(function (hit) {
                      result += text.substring(prevEnd, hit.position);
                      var end = hit.position + hit.length;
                      result += '<b class="search-keyword">' + text.substring(hit.position, end) + '</b>';
                      prevEnd = end;
                    });
                    result += text.substring(prevEnd, slice.end);
                    return result;
                  }

                  var resultItem = '';

                  if (slicesOfTitle.length != 0) {
                    resultItem += "<li><a href='" + articleUrl + "' class='search-result-title'>" + highlightKeyword(title, slicesOfTitle[0]) + "</a>";
                  } else {
                    resultItem += "<li><a href='" + articleUrl + "' class='search-result-title'>" + title + "</a>";
                  }

                  slicesOfContent.forEach(function (slice) {
                    resultItem += "<a href='" + articleUrl + "'>" +
                      "<p class=\"search-result\">" + highlightKeyword(content, slice) +
                      "...</p>" + "</a>";
                  });

                  resultItem += "</li>";
                  resultItems.push({
                    item: resultItem,
                    searchTextCount: searchTextCount,
                    hitCount: hitCount,
                    id: resultItems.length
                  });
                }
              })
            };
            if (keywords.length === 1 && keywords[0] === "") {
              resultContent.innerHTML = '<div id="no-result"><i class="fa fa-search fa-5x" /></div>'
            } else if (resultItems.length === 0) {
              resultContent.innerHTML = '<div id="no-result"><i class="fa fa-frown-o fa-5x" /></div>'
            } else {
              resultItems.sort(function (resultLeft, resultRight) {
                if (resultLeft.searchTextCount !== resultRight.searchTextCount) {
                  return resultRight.searchTextCount - resultLeft.searchTextCount;
                } else if (resultLeft.hitCount !== resultRight.hitCount) {
                  return resultRight.hitCount - resultLeft.hitCount;
                } else {
                  return resultRight.id - resultLeft.id;
                }
              });
              var searchResultList = '<ul class=\"search-result-list\">';
              resultItems.forEach(function (result) {
                searchResultList += result.item;
              })
              searchResultList += "</ul>";
              resultContent.innerHTML = searchResultList;
            }
          }

          if ('auto' === 'auto') {
            input.addEventListener('input', inputEventFunction);
          } else {
            $('.search-icon').click(inputEventFunction);
            input.addEventListener('keypress', function (event) {
              if (event.keyCode === 13) {
                inputEventFunction();
              }
            });
          }

          // remove loading animation
          $(".local-search-pop-overlay").remove();
          $('body').css('overflow', '');

          proceedsearch();
        }
      });
    }

    // handle and trigger popup window;
    $('.popup-trigger').click(function(e) {
      e.stopPropagation();
      if (isfetched === false) {
        searchFunc(path, 'local-search-input', 'local-search-result');
      } else {
        proceedsearch();
      };
    });

    $('.popup-btn-close').click(onPopupClose);
    $('.popup').click(function(e){
      e.stopPropagation();
    });
    $(document).on('keyup', function (event) {
      var shouldDismissSearchPopup = event.which === 27 &&
        $('.search-popup').is(':visible');
      if (shouldDismissSearchPopup) {
        onPopupClose();
      }
    });
  </script>





  

  

  

  

  

  

</body>
</html>
